Cost and Precision of Brownian Clocks

Brownian clocks are biomolecular networks that can count time. A paradigmatic example are proteins that go through a cycle, thus regulating some oscillatory behavior in a living system. Typically, such a cycle requires free energy often provided by ATP hydrolysis. We investigate the relation between the precision of such a clock and its thermodynamic costs. For clocks driven by a constant thermodynamic force, a given precision requires a minimal cost that diverges as the uncertainty of the clock vanishes. In marked contrast, we show that a clock driven by a periodic variation of an external protocol can achieve arbitrary precision at arbitrarily low cost. This result constitutes a fundamental difference between processes driven by a fixed thermodynamic force and those driven periodically. As a main technical tool, we map a periodically driven system with a deterministic protocol to one subject to an external protocol that changes in stochastic time intervals, which simplifies calculations significantly. In the nonequilibrium steady state of the resulting bipartite Markov process, the uncertainty of the clock can be deduced from the calculable dispersion of a corresponding current.

This article appears in the following collection:

Popular Summary

A crucial aspect of any clock is its precision. In the macroscopic world, clocks that have very high precision are the norm. But in the biological realm, clocks have to operate in a thermal environment, which exposes their constitutive parts to stochastic thermal noise, potentially affecting their precision. One might wonder whether for such “Brownian clocks” higher precision implies a higher energetic cost. Using the tools of stochastic thermodynamics, we investigate here the putative trade-off between precision and cost for two fundamentally different classes of Brownian clocks: clocks driven by a constant thermodynamic force and clocks driven by a periodic external energy source.

We model a Brownian clock as a random walk along a ring with $N$ states. For clocks driven by a constant thermodynamic force, such as the free energy liberated in ATP hydrolysis, there is indeed an inequality that establishes a minimal cost for a given precision. In contrast, for clocks driven by a time-periodic external protocol that modulates the energy of the states and the energy barriers between them, we show that by fine-tuning this energy landscape, one can indeed obtain a given precision with an arbitrarily low energy budget, provided the number of states is very large. Our findings provide a new perspective on biophysical and biochemical systems that are involved in “counting time.” We expect that future work may focus on determining how close to the theoretical limit real biological clocks operate and whether this trade-off has been a determining factor in their evolution.

Our results motivate the ongoing challenge of experimentally realizing a Brownian clock and will inform follow-up studies of the design of optimal artificial clocks operating on microscales and nanoscales.

Figure 1

Illustration of a Brownian clock with four states. The pointer in state 1 moves to state 2 with rate $k_1^{+}$, or to state 4 with rate $k_1^{-}$, and so on.

Figure 2

Illustration of the deterministic protocol for $N=4$. The red numbers 1, 2, 3, and 4 on the inner cycle denote the position in the clock, with the black arrow marking $i=1$ throughout. The energies $E_{\alpha }$ and the energy barriers $B_{\alpha }$ rotate one step in the clockwise direction after a time interval $\tau /N$, as indicated by the long green arrows. For the variable $\alpha$, these changes represented by the long green arrows correspond to an effective backward transition. For example, in the change from the top left to the top right that happens at time $t=\tau /4$, the variable $i=1$ remains fixed, and the variable $\alpha$ changes from $\alpha =1$ to $\alpha =4$.

Figure 3

Effective network for a clock driven by an external protocol that changes at stochastic times with $N=4$ states. The green backward arrows represent a jump with rate $\gamma =N/\tau$. A backward jump is equivalent to a forward rotation of the rates represented in Fig. 2.

Figure 4

Optimal profile $\{E_{\alpha }\}$ for which the product $\mathcal{C}{\epsilon }^2$ is minimized.

Figure 5

Product $\mathcal{C}{\epsilon }^2$ for a clock driven by an external protocol. The parameters of the clock are set to ${\chi }_1={\chi }_2=1$, ${\chi }_3={10}^{-B}$, $\gamma ={10}^{-x}$, $E_1=0$, $E_2=-1.21938$, and $E_3=-1.43550$. Below the lines, the product $\mathcal{C}{\epsilon }^2$ is smaller than $(\mathcal{A}/3)\coth (\mathcal{A}/6)$, which is the optimal value of this product for a clock driven by a fixed affinity $\mathcal{A}$ and $N=3$.

Figure 6

Comparison between the periodic steady state obtained with the continuous deterministic protocol from Eq. (a1) and the steady state obtained with the stochastic protocol that jumps with a rate $\gamma =L/\tau$. For the second case, the horizontal axis is $t=nT/L$. For the periodic steady state, we plot $R^{\mathrm{PS}}(t)$, and for the steady state, we plot the conditional probability $P(u|n)$ for different values of $L$. The parameters are set to $\omega =1$ and $E_0=1$.

Figure 7

Two-state model with a stochastic protocol. The states of the systems $d$ and $u$ have energies 0 and $E^n$, respectively. The protocol changes from $n$ to $n+1$ with a rate $\gamma$.

Figure 8

Rate of work done by the external process $\dot{w}$ as a function of $\omega =2\pi \gamma /N$. For the periodic steady state, this rate is denoted by ${\dot{w}}^{\mathrm{PS}}$.